Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Apply Newton's Laws to objects in equilibrium.
• Resolve forces into components and find the resultant force.
• Solve problems with connected objects, including on inclined planes.
• Calculate static and kinetic frictional forces.
• Define and explain tension in strings and ropes.

Lesson summary

Newton's Laws of Motion are fundamental to understanding how objects move and interact with each other. These laws govern everything from the motion of a car on a highway to the trajectory of a cricket ball. In South Africa, understanding these principles is crucial not only for academic success but also for various practical applications, such as engineering safer roads, designing efficient vehicles, and improving sporting performance. This week, we will focus on applying Newton's Laws to solve problems involving multiple forces and motion in two dimensions.

Lesson notes

Newton's Laws of Motion (Review): Newton's First Law (Law of Inertia):* An object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction unless acted upon by a net force. Inertia is the tendency of an object to resist changes in its state of motion.

Newton's Second Law:* The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Mathematically, this is expressed as F net = ma, where F net is the net force, m is the mass, and a is the acceleration. Remember, force and acceleration are vector quantities, meaning they have both magnitude and direction.

Newton's Third Law: For every action, there is an equal and opposite reaction. When one object exerts a force on another object, the second object exerts an equal and opposite force on the first object. These forces act on different objects.

Force Diagrams (Free-Body Diagrams): A force diagram is a visual representation of all the forces acting on an object. It's essential for applying Newton's Second Law correctly.

Here's how to construct one: Represent the object as a point or a simple shape. Draw vectors representing each force acting on the object. The length of the vector represents the magnitude of the force, and the direction of the vector represents the direction of the force. Label each force vector clearly.

Common forces include: Weight (W = mg, acting downwards) Normal force (N, acting perpendicular to the surface) Tension (T, acting along a string or rope) Applied force (F app , any force pushing or pulling the object) Friction (f, opposing motion)

Resolving Forces into Components: When forces act at angles, we need to resolve them into horizontal (x) and vertical (y) components. This makes it easier to calculate the net force in each direction. If a force F acts at an angle θ to the horizontal, its components are: F x = F cos θ F y = F sin θ Friction: Friction is a force that opposes motion between two surfaces in contact.

There are two types: Static Friction (f s ):* The force that prevents an object from starting to move.

It has a maximum value: f s,max = μ s N, where μ s is the coefficient of static friction and N is the normal force.

Kinetic Friction (f k ):* The force that opposes the motion of an object already in motion.

It is given by: f k = μ k N, where μ k is the coefficient of kinetic friction and N is the normal force. μ k is typically less than μ s .

Tension: Tension is the force transmitted through a string, rope, cable, or wire when it is pulled tight by forces acting from opposite ends. The tension force is directed along the length of the string and pulls equally on the objects on either end of the string.

Example 1: Object on an Inclined Plane A 5 kg block is placed on an inclined plane that makes an angle of 30° with the horizontal. The coefficient of kinetic friction between the block and the plane is 0.

2. Calculate the acceleration of the block down the plane. (Assume g = 9.8 m/s 2 )

Solution: Draw a free-body diagram: The forces acting on the block are weight (W), normal force (N), and friction (f k ).

Resolve the weight into components: W x = W sin θ = mg sin θ = (5 kg)(9.8 m/s 2 )sin 30° = 24.5 N (down the plane) W y = W cos θ = mg cos θ = (5 kg)(9.8 m/s 2 )cos 30° = 42.4 N (perpendicular to the plane)

Calculate the normal force: Since there is no acceleration perpendicular to the plane, N = W y = 42.4 N Calculate the kinetic friction: f k = μ k N = (0.2)(42.4 N) = 8.48 N (up the plane)

Apply Newton's Second Law along the plane: F net,x = W x - f k = ma 24.5 N - 8.48 N = (5 kg)a 16.02 N = (5 kg)a a = 3.20 m/s 2 (down the plane)

Example 2: Connected Objects Two blocks are connected by a light string that passes over a frictionless pulley. Block A has a mass of 3 kg and rests on a horizontal surface. Block B has a mass of 2 kg and hangs freely. The coefficient of kinetic friction between block A and the surface is 0.

3. Calculate the acceleration of the system and the tension in the string.